When the target class \(\mathcal {G}\)  is characterized by a finite family of forbidden induced subgraphs, modification problems are FPT by a result of Cai [1]}. Indeed, as long as the graph contains one of the forbidden subgraphs, one can try each possibility to correct this obstruction and branch by recursive calls. On each branch, the budget \(k\)  is strictly diminished, therefore the whole algorithm has a number of calls bounded by some function \(f(k)\) . The situation is more complicated when the target class \(\mathcal {G}\)  is characterized by an infinite family of forbidden induced subgraphs. Nonetheless, a large literature is devoted to edge modification problems towards chordal graphs (where we forbid all induced cycles with at least four vertices) as well as sub-classes of chordal graphs, typically obtained by requiring some fixed set of obstructions, besides the long cycles. Observe that, in this case, the situation remains relatively simple if we restrict ourselves to edge completion problems, where we are only allowed to add edges to the input graphs. Indeed, in this case, if a graph has an induced cycle of length longer than \(k+3\) , it cannot be made chordal by adding at most \(k\)  edges. Therefore we can use again the approach of Cai to deal with cycles of length at most \(k+3\)  and other obstructions, and either the algorithm finds a solution in \(f(k)\)  recursive calls, or we can conclude that we have a no-instance. The cases of edge deletion problems (where we are only allowed to remove edges) and edge editing problems (where we are allowed to both remove edges or add missing edges) are more complicated, since even long cycles can be eliminated by a single edge removal. Therefore more efforts and more sophisticated techniques were necessary in these situations, but several such problems turned out to be FPT [2]}, [3]}, [4]}.
The interested reader can refer to [5]} for a broad and comprehensive survey on parameterized algorithms for edge modification problems.
